Question

Simplify the expression below and explain which rules of exponents you used to simplify the expression. \frac{(16 \sqrt[]{x})^2 }{y^{-1}}

Answer 1

Given the following expression:

`\frac{(16 \sqrt[]{x})^2 }{y^(-1)}`

You need to remember the Rules of Exponents shown below:

- Negative Exponent Rule:

`b^(-n)=(1)/(b^n)`

Where "b" is the base and "n" is the exponent.

- Fractional Exponent Rule:

`b^{(m)/(n)}=\sqrt[n]{b^m}`

- Power of a Product Rule:

`(ab)^m=a^mb^m`

In this case, knowing the rules shown above, you can simplify the expression as follows:

1. Apply the Power of a Product Rule in the numerator:

`=\frac{(16)^2(\sqrt[]{x})^2}{y^(-1)}=\frac{256^{}(\sqrt[]{x})^2}{y^(-1)}`

2. Apply the Fractional Exponent Rule to simplify the square root:

`=\frac{256^{}(x^{(2)/(2)})^{}}{y^(-1)}=\frac{256^{}(x^1)^{}}{y^(-1)}=\frac{256^{}x^{}}{y^(-1)}`

3. Finally, apply the Negative Exponent Rule:

`=(256^{}x)(y^1)=256^{}xy`

Therefore, the answers are:

- Expression simplified:

`256^{}xy`

- Rules of Exponents used to simplify it:

1. Power of a Product Rule:

`(ab)^m=a^mb^m`

2. Fractional Exponent Rule:

`b^{(m)/(n)}=\sqrt[n]{b^m}`

3. Negative Exponent Rule:

`b^(-n)=(1)/(b^n)`